For a nonnegative function defined on an interval
define the set
(i.e., the region enclosed by the graph of the function and the -axis between the vertical lines at
and
). Then prove
where is the greatest integer less than or equal to
.
We can help ourselves by drawing a picture to get a good idea of what is going on, then turn that intuition into something more rigorous. The picture is as follows:
In the picture, we can see the number of lattice points in the ordinate set of (not including the
-axis since the question stipulates
contains the points
). At each integer between
and
, we count the number of lattice points beneath
, the greatest integer less than or equal to
. Then we need to turn this intuition from the picture into a proof:
Proof. Let with
. We know such an
exists since
with
. Then, the number of lattice points in
with first element
is the number of integers
such that
. But, by definition, this is
(the greatest integer less than or equal to
). Summing over all integers
we have,